| L(s) = 1 | − 1.53·2-s + 0.734·3-s + 0.370·4-s + 3.72·5-s − 1.13·6-s + 3.41·7-s + 2.50·8-s − 2.46·9-s − 5.73·10-s + 1.47·11-s + 0.271·12-s + 13-s − 5.25·14-s + 2.73·15-s − 4.60·16-s − 5.48·17-s + 3.78·18-s + 1.38·19-s + 1.37·20-s + 2.51·21-s − 2.26·22-s − 2.95·23-s + 1.84·24-s + 8.88·25-s − 1.53·26-s − 4.01·27-s + 1.26·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.424·3-s + 0.185·4-s + 1.66·5-s − 0.461·6-s + 1.29·7-s + 0.887·8-s − 0.820·9-s − 1.81·10-s + 0.443·11-s + 0.0784·12-s + 0.277·13-s − 1.40·14-s + 0.707·15-s − 1.15·16-s − 1.33·17-s + 0.892·18-s + 0.316·19-s + 0.308·20-s + 0.547·21-s − 0.482·22-s − 0.615·23-s + 0.376·24-s + 1.77·25-s − 0.301·26-s − 0.772·27-s + 0.238·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216990448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216990448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 - 0.734T + 3T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 + 2.95T + 23T^{2} \) |
| 29 | \( 1 - 9.19T + 29T^{2} \) |
| 37 | \( 1 + 6.83T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 + 0.260T + 43T^{2} \) |
| 47 | \( 1 - 9.41T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 - 4.52T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 7.02T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 8.44T + 79T^{2} \) |
| 83 | \( 1 + 2.77T + 83T^{2} \) |
| 89 | \( 1 - 1.47T + 89T^{2} \) |
| 97 | \( 1 - 2.61T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.85011297742015103142196232002, −10.26531975815669809453886779532, −9.101206014905146786681173629196, −8.841957346500900750940719172529, −7.957281974755866168267714949627, −6.67425587402125532667926806193, −5.55906433555395282007025602755, −4.49571181976976111619335868229, −2.41393724248131627596301550516, −1.47699257426153556139089251548,
1.47699257426153556139089251548, 2.41393724248131627596301550516, 4.49571181976976111619335868229, 5.55906433555395282007025602755, 6.67425587402125532667926806193, 7.957281974755866168267714949627, 8.841957346500900750940719172529, 9.101206014905146786681173629196, 10.26531975815669809453886779532, 10.85011297742015103142196232002
